Method of determination of fluid influx profile and near-wellbore space parameters

ABSTRACT

Method for determination of a fluid influx profile and near-wellbore area parameters comprises measuring a first bottomhole pressure and after operating a well at a constant production rate changing the production rate and measuring a second bottomhole pressure. A wellbore fluid temperature over an upper boundary of a lowest productive layer and wellbore fluid temperatures above and below other productive layers are measured and relative production rates and skin factors of the productive layers are calculated from measured wellbore fluid temperatures and measured first and second bottomhole pressures.

FIELD OF THE DISCLOSURE

The invention relates to the area of geophysical studies of oil and gas wells, particularly, to the determination of the fluid influx profile and multi-layered reservoir near-wellbore area space parameters.

BACKGROUND OF THE DISCLOSURE

A method to determine relative production rates of the productive layers using quasi-steady flux temperature values measured along the wellbore is described, e.g. in: {hacek over (C)}eremenskij G.A. Prikladnaja geotermija, Nedra, 1977 p. 181. A disadvantage of this method is a low accuracy of the layers' relative flow rate determination resulting from the assumption of the Joule-Thomson effect constant value for different layers. In effect, it depends on the formation pressure and specific layer pressure values.

SUMMARY OF THE DISCLOSURE

The technical result of the invention is an increased accuracy of the wellbore parameters (influx profile, values of skin factors for different productive layers) determination.

The claimed method comprises the following steps. A bottomhole pressure is measured; after a long-term operation of the well at a constant production rate during the time sufficient to provide a minimum influence of the production time on the rate of the subsequent change of the temperature of the fluids flowing from the production layers into the wellbore, the production rate is changed. After change the bottomhole pressure and the wellbore fluid temperature near an upper boundary of the lowest productive layer as well as above and below the other productive layers are measured. The graphs of the dependence of the temperature measured over the lowest layer as function of time and the derivative of this temperature by the logarithm of the time elapsed after the wellbore production rate change are plotted. A skin factor of the lowest layer is determined by the graphs obtained. Temperatures of the fluids flowing into the wellbore from the overlying layers are determined by iterative procedure using the measured temperatures, and relative production rates and skin factors of the overlying layers are calculated.

The total number of layers n in the method claimed is not limited. Particular distance from the temperature transmitters to the layers' boundaries shall be determined depending on the casing string diameter and wellbore production rate. In most cases the optimum distance is 1-2 meters. Processing of the data obtained using the method claimed in the invention enables finding production rates and skin factors of separate layers in the multi-layer wellbore.

BRIEF DESCRIPTION OF THE FIGURES

FIG. 1 shows the influence of the production time on the temperature change rate after the wellbore production rate has been changed;

FIG. 2 shows the dependencies of the influx temperature derivative dT_(in1)/dlnt and temperature measured over the first productive layer dT₀/dlnt vs. time;

FIG. 3 shows the dependencies of the influx temperature derivative dT_(in2)/dlnt and respective temperature calculated using an iterative procedure as a function of time;

FIG. 4 shows the temperature measured over the first productive layer and temperature of the influx from the second layer calculated using an iterative procedure as well as the determination of changes in influx temperatures ΔT_(d1) and ΔT_(d2) (at t_(d1) and t_(d2) time moments) and calculating the layers' skin factors; and

FIG. 5 shows the dependence of the bottomhole pressure as a function of time after the production rate change.

DETAILED DESCRIPTION

The method claimed in the invention is based on a simplified model of heat- and mass-exchange processes in the productive layer and wellbore. Let us consider the results of the model application for the processing of the measurement results of the temperature T_(in) ^((i))(t) of fluids flowing into the wellbore from two productive layers.

In the approximation of the productive layers' pressure stabilization, the change rate in the temperature of the fluid flowing into the wellbore after the production rate has been changed is described by Equation (1):

$\begin{matrix} {{\frac{T_{in}}{t} = {\frac{ɛ_{0}}{2 \cdot \left( {s + \theta} \right)} \cdot \left\lbrack {{\frac{P_{e} - P_{1}}{f\left( {t,t_{d\; 1}} \right)} \cdot \frac{1}{\left( {{\delta_{12} \cdot t_{p}} + t_{2} + t} \right)}} + {\frac{P_{1} - P_{2}}{f\left( {t,t_{d}} \right)} \cdot \frac{1}{\left( {t_{2} + t} \right)}}} \right\rbrack}},} & (1) \end{matrix}$

where P_(e) is a layer pressure, P₁ and P₂—bottomhole pressures before and after the change in the production rate, s—a layer skin factor, θ=ln(r_(e)/r_(w)), r_(e)—a drain radius, r_(w)—the wellbore radius, t—time counted from the moment of the change in the production rate, t_(p)—production time at the bottomhole pressure of

$\begin{matrix} {P_{1},{\delta_{12} = \frac{P_{e} - P_{1}}{P_{e} - P_{2}}},{{f\left( {t,t_{d}} \right)} = \left\{ {{\begin{matrix} K & {t \leq t_{d}} \\ 1 & {{t_{d} < t},} \end{matrix}K} = {\frac{k_{d}}{k} = \left\lbrack {1 + \frac{s}{\theta_{d}}} \right\rbrack^{- 1}}} \right.}} & (2) \end{matrix}$

—a relative permeability of the bottom-hole zone, θ_(d)=ln(r_(d)/r_(w)), r_(d)—bottom-hole zone radius, t_(d1)=t₁·D and t_(d2)=t₂·D—certain characteristic heat-exchange times in layer 1 and layer 2, D=(r_(d)/r_(w))²−1—non-dimensional parameter characterizing the size of the near-wellbore area,

${t_{1,2} = \frac{\pi \cdot r_{w}^{2}}{\chi \cdot q_{1,2}}},{q_{1,2} = {\frac{Q_{1,2}}{h} = {\frac{2{\pi \cdot k}}{\mu} \cdot \frac{\left( {P_{e} - P_{1,2}} \right)}{s + \theta}}}}$

—specific volumetric production rates before (index 1) and after (index 2) the production rate change, Q_(1,2), h and k—volumetric production rates, the thickness and permeability of a layer,

${\chi = \frac{c_{f} \cdot \rho_{f}}{\rho_{r} \cdot c_{r}}},{{\rho_{r}c_{r}} = {{{\varphi \cdot \rho_{f}}c_{f}} + {{\left( {1 - \varphi} \right) \cdot \rho_{m}}c_{m}}}},$

φ is a layer's porosity, ρ_(f)c_(f)—volumetric heat capacity of the fluid, ρ_(m)c_(m)—volumetric heat capacity of the rock matrix, μ—fluid viscosity. r_(d)—external radius of the near-wellbore zone with the permeability and fluid influx profile changed as compared with the properties of a layer far away from the wellbore (to be determined by a set of factors, like perforation holes' properties, permeability distribution in the affected zone around the wellbore and drilling incompleteness).

According to Equation (1) at a relatively long production time t_(p) before the production rate has been changed its influence on the temperature change dynamics tends towards zero. Let us quantify this influence. For the order of magnitude χ≈0.7, r_(w)≈0.1 m, and for r_(d)≈0.3 m q=100 [m³/day]/3 m≈4.10⁻⁴ m³/s we have: t₂≈0.03 hours, t_(d)≈0.25 hours. If the measurement time t is t≈2÷3 hours (i.e. t>>t₂, t_(d) and f (t,t_(d))=1) it is possible to evaluate what relative error is introduced into the derivative (1) value by the final production time before the measurements:

$\begin{matrix} {{\frac{1}{{\overset{.}{T}}_{in}} \cdot {\Delta \left( {\overset{.}{T}}_{in} \right)}} = {\frac{P_{e} - P_{1}}{P_{1} - P_{2}} \cdot \frac{1}{1 + \frac{t_{p}}{t}}}} & (3) \end{matrix}$

FIG. 1 shows the calculation results using Equation (3) for P_(e)=100 Bar, P₁=50 Bar, P₂=40 Bar and t_(p)=5, 10 and 30 days. From the Figure we can see, for example, that if the time of production at a constant production rate was 10 or more days, then within the time t=3 hours after the production rate has changed the influence of t_(p) value on the influx temperature change rate will not exceed 6%. It is essential that an increase in the measurement time t results in a proportional increase in the required production time at the constant production rate before the measurements, so that the error introduced into the derivative (1) by the value t_(p) could be maintained unchanged.

Then it is assumed that the production time t_(p) is long enough and Equation (1) may be written as:

$\begin{matrix} {\frac{T_{i\; n}}{t} \approx {\frac{ɛ_{0} \cdot \left( {P_{1} - P_{2}} \right)}{2 \cdot \left( {s + \theta} \right)} \cdot \frac{1}{f\left( {t,t_{d}} \right)} \cdot \frac{1}{t}}} & (4) \end{matrix}$

From Equation (4) it is seen that at long enough time values t>t_(d), where

$\begin{matrix} {t_{d} = \frac{\pi \cdot r_{w}^{2} \cdot D}{\chi \cdot q_{2}}} & (5) \end{matrix}$

the temperature change rate as function of time is described as a simple proportion:

$\frac{T_{i\; n}}{{\ln}\; t} = {{const}.}$

Numerical modeling of the heat- and mass-exchange processes in the productive layers and production wellbore shows that the moment t=t_(d) may be singled out at the graph of

$\frac{T_{i\; n}}{{\ln}\; t}$

vs. time as the start of the logarithmic derivative constant value section.

If we assume that the dimensions of the bottomhole areas in different layers are approximately equal (D₁≈D₂), then using times t_(d) ⁽¹⁾ and t_(d) ⁽²⁾, found for two different layers their relative production rates may be found using Equation (6).

$Y = \frac{q_{2}h_{2}}{{q_{1}h_{1}} + {q_{2}h_{2}}}$ or $Y = {\left( {1 + \frac{q_{1} \cdot h_{1}}{q_{2} \cdot h_{2}}} \right)^{- 1} = \left( {1 + {\frac{h_{1}}{t_{d\;}^{(1)}} \cdot \frac{t_{d}^{(2)}}{h_{2}}}} \right)^{- 1}}$

In general relative production rates of the second, third etc. layers is calculated using Equation (6):

$\begin{matrix} {{{Y_{2} = {\frac{q_{2}h_{2}}{{q_{1}h_{1}} + {q_{2}h_{2}}} = \left\lbrack {1 + {\left( \frac{h_{1}}{t_{d,1}} \right) \cdot \frac{t_{d,2}}{h_{2}}}} \right\rbrack^{- 1}}},{Y_{3} = {\frac{q_{3}h_{3}}{{q_{1}h_{1}} + {q_{2}h_{2}} + {q_{3}h_{3}}} = \left\lbrack {1 + {\left( {\frac{h_{1}}{t_{d\;,1}} + \frac{h_{2}}{t_{d,2}}} \right) \cdot \frac{t_{d,3}}{h_{3}}}} \right\rbrack^{- 1}}},\begin{matrix} {Y_{4} = \frac{q_{4}h_{4}}{{q_{1}h_{1}} + {q_{2}h_{2}} + {q_{3}h_{3}} + {q_{4}h_{4}}}} \\ {{= \left\lbrack {1 + {\left( {\frac{h_{1}}{t_{d,1}} + \frac{h_{2}}{t_{d,2}} + \frac{h_{3}}{t_{d,3}}} \right) \cdot \frac{t_{d,4}}{h_{4}}}} \right\rbrack^{- 1}},} \end{matrix}}{{etc}.}} & (6) \end{matrix}$

Equation (1) is obtained for the cylindrically symmetrical flow in the layer and bottomhole area (with the bottomhole area permeability of k_(d)≠k), which has the external radius r_(d). The temperature distribution nature in the bottomhole area is different from the temperature distribution away from the wellbore. After the production rate has been changed this temperature distribution is carried over into the well by the fluid flow which results in the fact that the nature of T_(in)(t) dependence at low times (after the flow rate change) differs from T_(in)(t) dependence observed at large (t>t_(d)) time values. From Equation (7) it is seen that with the accuracy to χ coefficient the volume of the fluid produced required for the transition to the new nature of the dependence of the incoming fluid temperature T_(in)(t) vs, time is determined by the volume of the bottomhole area:

$\begin{matrix} {{t_{d} \cdot q_{2}} = {\frac{1}{\chi} \cdot \pi \cdot \left( {r_{d}^{2} - r_{w}^{2}} \right)}} & (7) \end{matrix}$

In case of perforated wellbore there always is a “bottomhole” area (regardless of the permeability's′ distribution) in which the temperature distribution nature is different from the temperature distribution in the layer away from the wellbore. This is the area where the fluid flow is not symmetrical and the size of this area is determined by the perforation tunnels' length (L_(p)):

$\begin{matrix} {D_{p} \approx {\left( \frac{r_{w} + L_{p}}{r_{w}} \right)^{2} - 1.}} & (8) \end{matrix}$

If we assume that the lengths of perforation tunnels in different productive layers are approximately equal (D_(p1)≈D_(p2)), then relative production rates of the layers are also determined by Equation (6). Equation (8) may be updated by introducing a numerical coefficient of about 1.5-2.0, the value of which may be determined from the comparison with the numerical calculations or field data.

To determine the layer skin factor s temperature difference ΔT_(d) of the fluid flowing into the wellbore during the time between the flow rate change and t_(d): time.

$\begin{matrix} {{\Delta \; T_{d}} = {\int_{0}^{t_{d}}{\frac{T_{i\; n}}{t} \cdot {{t}.}}}} & (9) \end{matrix}$

Using Equation (4) we find:

$\begin{matrix} {{{\Delta \; T_{d}} = {c \cdot ɛ_{0} \cdot \left( {P_{1} - P_{2}} \right) \cdot \frac{s + \theta_{d}}{s + \theta}}},} & (10) \end{matrix}$

where ΔT_(d) is the change of the influx temperature by the time t=t_(d), (P₁−P₂)—steady-state difference between the old and the new bottomhole pressure which is achieved in the wellbore several hours after the wellbore production rate has been changed. Whereas Equation (4) does not consider the influence of the end layer pressure field tuning rate, Equation (10) includes non-dimensional coefficient c (approximately equal to one) the value of which is updated by comparing with the numerical modeling results.

According to (10), skin factor s value is calculated using:

$\begin{matrix} {{s = \frac{{\psi \cdot \theta} - \theta_{d}}{1 - \psi}}{{{where}\mspace{14mu} \psi} = \frac{\Delta \; T_{d}}{c \cdot ɛ_{0} \cdot \left( {P_{1} - P_{2}} \right)}}} & (11) \end{matrix}$

When it is impossible to directly measure T_(in) ^((i))(t) (i=1, 2, . . . , n) of the fluids flowing into the wellbore from different layers we suggest using wellbore temperature measurement data and the following wellbore measurement data processing procedure.

Temperature T₀(t) measure near the upper boundary of the lower productive layer is (with a good accuracy) equal to the relevant influx temperature therefore using change rate T₀ the value of t_(d) ⁽¹⁾ is determined, influx temperature change is determined by the time ΔT(t_(d) ⁽¹⁾)=ΔT_(d) ⁽¹⁾ and using Equation (11) skin factor s₁ of the lower productive layer is found.

Relative production rate Y⁽²⁾ (Y⁽²⁾=Q₂/(Q₁+Q₂)) and skin factor of the second productive layer is found using the following iterative procedure. The arbitrary value of Y⁽²⁾ is set and using Equation (12):

$\begin{matrix} {{T_{i\; n}^{(2)}(t)} = {\frac{1}{Y^{(2)}} \cdot \left\lbrack {{T_{2}^{(2)}(t)} - {\left( {1 - Y^{(2)}} \right) \cdot {T_{1}^{(2)}(t)}}} \right\rbrack}} & (12) \end{matrix}$

the first approximation for the temperature of the fluid flowing into the wellbore from the second productive layer is found. Then, from the dependence T_(in) ⁽²⁾(t) t_(d) ⁽²⁾ is found and using Equation (6) the new value of relative production rate Y_(n) ⁽²⁾ is found:

$\begin{matrix} {Y^{(2)} = \left( {1 + {\frac{h_{1}}{t_{d}^{(1)}} \cdot \frac{t_{d}^{(2)}}{h_{2}}}} \right)^{- 1}} & (13) \end{matrix}$

If this value differs from Y⁽²⁾, the calculation using Equations (12) and (13) is repeated until these values are equal.

The Y⁽²⁾ value found is the relative production rate of the second layer and the respective t_(d) ⁽²⁾ value—the time of the influx from the bottomhole area for the second layer. Using the value Y⁽²⁾ from Equation (12) temperature T_(in) ⁽²⁾(t) of the influx from the second layer is found and using T_(in) ⁽²⁾(t) and the found t_(d) ⁽²⁾ value ΔT_(d) ⁽²⁾ is determined and using Equation (10) skin factor s₂ of the second layer is calculated.

The relative production rates Y^((i)) (Y^((i))=Q_(i)/(Q₁+Q₂+ . . . +Q_(i))) and skin factors of the overlying layers (i=2, 3 etc.) are determined subsequently starting from the second (from the bottom) layer using the following iterative procedure:

$\begin{matrix} {{{Set}\mspace{14mu} Y^{(i)}}{calculate}{{T_{i\; n}^{(i)}(t)} = {\frac{1}{Y^{(i)}} \cdot \left\lbrack {{T_{2}^{(i)}(t)} - {\left( {1 - Y^{(i)}} \right) \cdot {T_{1}^{(i)}(t)}}} \right\rbrack}}} & (14) \end{matrix}$

And by the dependence obtained we find the time t_(d) ^((i)) of the influx from the bottomhole area and calculate the new value of Y^((i)) using one of the equations below (depending on the layer number i), using the values of characteristic times t_(d) ^((i)), found for the layers below

$\begin{matrix} {{i = {2\text{:}}}{Y^{(2)} = \left\lbrack {1 + {\left( \frac{h_{1}}{t_{d}^{(1)}} \right) \cdot \frac{t_{d}^{(2)}}{h_{2}}}} \right\rbrack^{- 1}}{i = {3\text{:}}}{Y^{(3)} = \left\lbrack {1 + {\left( {\frac{h_{1}}{t_{d}^{(1)}} + \frac{h_{2}}{t_{d}^{(2)}}} \right) \cdot \frac{t_{d}^{(3)}}{h_{3}}}} \right\rbrack^{- 1}}{i = {4\text{:}}}{Y^{(4)} = \left\lbrack {1 + {\left( {\frac{h_{1}}{t_{d}^{(1)}} + \frac{h_{2}}{t_{d}^{(2)}} + \frac{h_{3}}{t_{d}^{(3)}}} \right) \cdot \frac{t_{d}^{(4)}}{h_{4}}}} \right\rbrack^{- 1}}{{etc}.}} & (15) \end{matrix}$

Therefore the determination of the influx profile and productive layers' skin factors by the results of transition processes' temperature measurement includes the following steps:

1. The well is operated at a constant production rate for a long time (from 5 to 30 days depending on the planned duration and measurement accuracy requirements).

2. Wellbore production rate is changed, hereby the bottomhole pressure and wellbore fluid temperature T₀(t) in the influx lower area as well as temperature values below and above the productive layers in question are measured.

3. Dependence of the logarithmic derivative dT₀/dlnt as function of time is measured and from this dependence curve t_(d) ⁽¹⁾, ΔT_(d) ⁽¹⁾ value is found and using Equation (11) skin factor s₁ of the lower layer is found.

4. Relative production rates and skin factors of the overlying layers (from i=2 to i=n) are found using iterative procedure (14)-(15).

The possibility of determination of the influx profile and productive layers' skin factors using the method claimed was checked on synthetic examples prepared using production wellbore numerical simulator which models non-steady pressure field in the wellbore-layers system, non-isothermal flow of the fluids compressed in a non-uniform porous medium, flow mixture in the wellbore and wellbore-layer heat-exchange etc.

FIG. 2-5 shows the results of the calculation for the following two-layer model:

k₁=100 mD, s₁=0.5, h₁=4 m

k₂=500 mD, s₂=7, h₂=6 m

The time of the production at a production rate of Q₁=300 m³/day is t_(p)=2000 hours; Q₂=400 m³/day. From FIG. 5 it is seen than in the case in question the wellbore pressure continues to significantly change even after 24 hours. FIG. 2 shows the dependences of the influx temperature dT_(in1)/dlnt derivative (solid line) and temperature measured over the first productive layer, dT₀/dlnt (dashed line) as function of time. FIG. 3 shows the dependences of the influx temperature dT_(in2)/dlnt derivative (solid line) and respective temperature calculated using iterative procedure (dashed line) as function of time. From these figures we can see that temperature T₀, and temperature of the influx from the upper layer obtained as a result of the iterative procedure yield the same values of characteristic times as the influx temperatures: t_(d) ⁽¹⁾=0.5 hours and t_(d) ⁽²⁾=0.3 hours. Using these values we find relative production rate of the upper layer 0.72 which is close to the true value (0.77). FIG. 4 shows temperature measured over the first productive layer and temperature of the influx from the second layer calculated using the iterative procedure. By the time moments t_(d1) and t_(d2) these temperatures' change is: ΔT_(d) ⁽¹⁾=0.098 K, ΔT_(d) ⁽²⁾=0.169 K. If in Equation (11) we assume the non-dimensional constant value of c=1.1, then the layers' skin factors calculated using these values will be different from the true values by maximum 20%. 

1. A method for the determination of a fluid influx profile and near-wellbore area parameters comprising: measuring a bottomhole pressure, operating a well at a constant production rate during a time sufficient to provide a minimum influence of a production time on a rate of a subsequent change of a temperature of the fluids flowing from production layers into a wellbore, changing the production rate, measuring the bottomhole pressure, measuring the wellbore fluid temperature near an upper boundary of a lowest productive layer as well as above and below other productive layers, plotting a graph of a dependence of the temperature measured over the lowest productive layer as a function of time and a derivative of this temperature by a logarithm of the time elapsed after the wellbore production rate has been changed, determining a skin factor of the lowest layer by the graph obtained, determining temperatures of the fluids flowing into the wellbore from overlying layers by iterative procedure using the measured temperatures, and calculating relative production rates and skin factors of the overlying layers. 